Finding the minimum of functions is crucial to many problems in ranging from astrophysics to quantum optimal design. Unfortunately, not always are these functions cheap-to-evaluate, which means the minimum has to be found by making as few observations as possible.To tackle this problem, we use an auxiliary function (expressed as a Gaussian process) to mimic the original function as observations are made. This function is relatively cheap-to-evaluate, and more importantly, has analytic derivatives. Therefore, it is possible to find the minimum of this auxiliary function using textbook gradient-based minimisation algorithms. If an enough number of observations have been made, the minimum of this auxiliary function is the minimum of the original function. As an application of this algorithm in physics, we find a laser pulse envelope which, when shone on a hydrogen atom, maximises the ionisation.